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こうしょえいさか
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2 1 (4/6) : ruby 2 / 35

3 ( ) : gnuplot 3 / 35

4 ( ) 4 / 35

5 (summary statistics) : (mean) (median) (mode) : (range) (variance) (standard deviation) 5 / 35

6 (mean): x = 1 n (median): { xr+1 m, m = 2r + 1 x median = (x r + x r+1 )/2 m, m = 2r n i=1 (mode): x i f(x) mode median mean median mode mean x 6 / 35

7 (percentiles) pth-percentile: p% median = 5th-percentile total observations (%) sorted variable x 7 / 35

8 (range): (variance): σ 2 = 1 nx (x i x) 2 n i=1 (standatd deviation): σ 68% (mean ± stddev) 95% (mean ± 2stddev) f(x) 1 mean median exp(-x**2/2) % x 95% 8 / 35

9 (variance): σ 2 = 1 nx (x i x) 2 n i=1 σ 2 = 1 nx (x i x) 2 n i=1 = 1 nx (xi 2 2x i x + x 2 ) n i=1 = 1 nx nx n ( xi 2 2 x x i + n x 2 ) i=1 i=1 = 1 nx xi 2 2 x 2 + x 2 n i=1 = 1 nx xi 2 x 2 n i=1 9 / 35

10 normalized traffic volume cdf time (sec) normalized traffic volume / 35

11 : sample data from a book: P. K. Janert Gnuplot in Action # Minutes Count :2,355 :171.3 :14.1 : / 35

12 : (2) count finish time (minutes) 12 / 35

13 : (3) rank finish time (minutes) 13 / 35

14 XY XY : ( ) XY 3D ( : ) 14 / 35

15 15 / 35

16 X Y 4 normalized traffic volume time (sec) 16 / 35

17 (1/2) X : Y : frequency normalized traffic volume 17 / 35

18 (2/2) ( ) ( ) 18 / 35

19 (probability density function; pdf) 1 : X x f (x) = P[X = x] pdf normalized traffic volume 19 / 35

20 (cumulative distribution function; cdf) : x f (x) = P[X = x] : x F (x) = P[X <= x] cdf normalized traffic volume 2 / 35

21 CDF CDF CDF 18 ping rtt 18 ping rtt histogram 1 8 histogram response time (msec) response time (msec) CDF samples 1 samples response time (msec) ( ) ( )1 ( )CDF 21 / 35

22 (scatter plots) 2 X : X Y : Y X Y : ( ).7 ( ). ( ) / 35

23 gnuplot grace GUI 23 / 35

24 : : P. K. Janert Gnuplot in Action 24 / 35

25 : ( ) # regular expression to read minutes and count re = /^(\d+)\s+(\d+)/ sum = # sum of data n = # the number of data ARGF.each_line do line if re.match(line) min = $1.to_i cnt = $2.to_i sum += min * cnt n += cnt end end mean = Float(sum) / n printf "n:%d mean:%.1f\n", n, mean % ruby mean.rb marathon.txt n:2355 mean: / 35

26 : : σ 2 = 1 n n i=1 (x i x) 2 # regular expression to read minutes and count re = /^(\d+)\s+(\d+)/ data = Array.new sum = # sum of data n = # the number of data ARGF.each_line do line if re.match(line) min = $1.to_i cnt = $2.to_i sum += min * cnt n += cnt for i in 1.. cnt data.push min end end end mean = Float(sum) / n sqsum =. data.each do i sqsum += (i - mean)**2 end var = sqsum / n stddev = Math.sqrt(var) printf "n:%d mean:%.1f variance:%.1f stddev:%.1f\n", n, mean, var, stddev % ruby stddev.rb marathon.txt n:2355 mean:171.3 variance:199.9 stddev: / 35

27 : : σ 2 = 1 n n i=1 x 2 i x 2 # regular expression to read minutes and count re = /^(\d+)\s+(\d+)/ sum = # sum of data n = # the number of data sqsum = # su of squares ARGF.each_line do line if re.match(line) min = $1.to_i cnt = $2.to_i sum += min * cnt n += cnt sqsum += min**2 * cnt end end mean = Float(sum) / n var = Float(sqsum) / n - mean**2 stddev = Math.sqrt(var) printf "n:%d mean:%.1f variance:%.1f stddev:%.1f\n", n, mean, var, stddev % ruby stddev2.rb marathon.txt n:2355 mean:171.3 variance:199.9 stddev: / 35

28 : # regular expression to read minutes and count re = /^(\d+)\s+(\d+)/ data = Array.new ARGF.each_line do line if re.match(line) min = $1.to_i cnt = $2.to_i for i in 1.. cnt data.push min end end end data.sort! # just in case data is not sorted n = data.length # number of array elements r = n / 2 # when n is odd, n/2 is rounded down if n % 2!= median = data[r] else median = (data[r - 1] + data[r])/2 end printf "r:%d median:%d\n", r, median % ruby median.rb marathon.txt r:1177 median: / 35

29 : gnuplot gnuplot 29 / 35

30 plot "marathon.txt" using 1:2 with boxes ( ) set boxwidth 1 set xlabel "finish time (minutes)" set ylabel "count" set yrange [:18] set grid y plot "marathon.txt" using 1:2 with boxes notitle "marathon.txt" using 1:2 count finish time (minutes) 3 / 35

31 : CDF : # Minutes Count : # Minutes Count CumulativeCount / 35

32 : CDF (2) ruby code: re = /^(\d+)\s+(\d+)/ cum = ARGF.each_line do line begin if re.match(line) # matched time, cnt = $~.captures cum += cnt.to_i puts "#{time}\t#{cnt}\t#{cum}" end end end gnuplot command: set boxwidth 1 set xlabel "finish time (minutes)" set ylabel "CDF" set grid y plot "marathon-cdf.txt" using 1:($3 / 2355) with lines notitle 32 / 35

33 CDF CDF finish time (minutes) 33 / 35

34 ( ) : gnuplot 34 / 35

35 3 (4/2) : 35 / 35

¥¤¥ó¥¿¡¼¥Í¥Ã¥È·×Â¬¤È¥Ç¡¼¥¿²òÀÏ Âè2²ó

¥¤¥ó¥¿¡¼¥Í¥Ã¥È·×Â¬¤È¥Ç¡¼¥¿²òÀÏ Âè2²ó 2 2015 4 20 1 (4/13) : ruby 2 / 49 2 ( ) : gnuplot 3 / 49 1 1 2014 6 IIJ / 4 / 49 1 ( ) / 5 / 49 ( ) 6 / 49 (summary statistics) : (mean) (median) (mode) : (range) (variance) (standard deviation) 7 / 49